2-连通坚韧图中的长圈

中文摘要 AI 辅助

设$G$为一个图。$G$的周长，记作$cir(G)$，是$G$中最长圈的长度，若$G$无圈则为零。1993年，Broersma、van den Heuvel、Jung和Veldman猜想：对每个$t>0$，存在常数$A=A(t)>0$，使得每个$n$阶2-连通$t$-坚韧图的周长至少为$A\log n$；该猜想被记录为Bauer、Broersma和Schmeichel在2006年关于坚韧度的综述中的猜想2。在本文中，我们证实了这一猜想。更精确地，每个$n$阶2-连通$t$-坚韧图$G$满足$cir(G)\ge \lceil \log_k((k-1)n+1)\rceil$，其中$k=\lceil 1/t\rceil+2$。证明结合了Win的有界度生成树定理与Briański、Joret、Majewski、Micek、Seweryn和Sharma的定理：2-连通图的树深度至多为其周长。

英文摘要

Let $G$ be a graph. The circumference of $G$, denoted by $cir(G)$, is the length of a longest cycle in $G$, or zero if $G$ is acyclic. In 1993, Broersma, van den Heuvel, Jung, and Veldman conjectured that, for every $t>0$, there is a constant $A=A(t)>0$ such that every 2-connected $t$-tough graph of order $n$ has circumference at least $A\log n$; the conjecture is recorded as Conjecture~2 in the 2006 survey on toughness by Bauer, Broersma, and Schmeichel. In this note, we confirm the conjecture. More precisely, every 2-connected $t$-tough graph $G$ of order $n$ satisfies $cir(G)\ge \lceil \log_k((k-1)n+1)\rceil$, where $k=\lceil 1/t\rceil+2$. The proof combines Win's bounded-degree spanning tree theorem with the theorem of Briański, Joret, Majewski, Micek, Seweryn, and Sharma that the treedepth of a 2-connected graph is at most its circumference.

发表机构

• Auburn University（奥本大学）

机构由 AI 辅助整理，请以论文原文为准。